Two particles P and Q have mass 0.4 kg and 0.6 kg respectively. The particles are initially at rest on a smooth horizontal table. Particle P is given an impulse of magnitude 3 N s in the direction PQ.
Find the speed of P immediately before it collides with Q.
(3)
Immediately after the collision between P and Q, the speed of Q is 5 m s–1.
Show that immediately after the collision P is at rest.
(3)
At time t = 0, a particle is projected vertically upwards with speed u m s–1 from a point 10 m above the ground. At time T seconds, the particle hits the ground with speed 17.5 m s–1. Find
A particle P of mass 0.4 kg moves under the action of a single constant force F newtons. The acceleration of P is (6i + 8j) m s–2. Find
the angle between the acceleration and i,
(2)
the magnitude of F.
(3)
At time t seconds the velocity of P is v m s–1. Given that when t = 0, v = 9i – 10j,
find the velocity of P when t = 5.
(3)
A car is moving along a straight horizontal road. The speed of the car as it passes the point A is 25 m s–1 and the car maintains this speed for 30 s. The car then decelerates uniformly to a speed of 10 m s–1. The speed of 10 m s–1 is then maintained until the car passes the point B. The time taken to travel from A to B is 90 s and AB = 1410 m.
Sketch, in the space below, a speed-time graph to show the motion of the car from A to B.
(2)
Calculate the deceleration of the car as it decelerates from 25 m s–1 to 10 m s–1.
(7)
Two forces P and Q act on a particle at a point O. The force P has magnitude 15 N and the force Q has magnitude X newtons. The angle between P and Q is 150°, as shown in Figure 1. The resultant of P and Q is R. Given that the angle between R and Q is 50°, find
the magnitude of R,
(4)
the value of X.
(5)
A plank AB has mass 12 kg and length 2.4 m. A load of mass 8 kg is attached to the plank at the point C, where AC = 0.8 m. The loaded plank is held in equilibrium, with AB horizontal, by two vertical ropes, one attached at A and the other attached at B, as shown in Figure 2. The plank is modelled as a uniform rod, the load as a particle and the ropes as light inextensible strings.
The plank is now modelled as a non-uniform rod. With the new model, the tension in the rope attached at A is 10 N greater than the tension in the rope attached at B.
Find the distance of the centre of mass of the plank from A.
(6)
A package of mass 4 kg lies on a rough plane inclined at 30° to the horizontal. The package is held in equilibrium by a force of magnitude 45 N acting at an angle of 50° to the plane, as shown in Figure 3. The force is acting in a vertical plane through a line of greatest slope of the plane. The package is in equilibrium on the point of moving up the plane. The package is modelled as a particle. Find
the magnitude of the normal reaction of the plane on the package,
(5)
the coefficient of friction between the plane and the package.
(6)
Two particles P and Q, of mass 2 kg and 3 kg respectively, are joined by a light inextensible string. Initially the particles are at rest on a rough horizontal plane with the string taut. A constant force F of magnitude 30 N is applied to Q in the direction PQ, as shown in Figure 4. The force is applied for 3 s and during this time Q travels a distance of 6 m. The coefficient of friction between each particle and the plane is 
. Find
State how in your calculation you have used the information that the string is inextensible.
(1)
When the particles have moved for 3 s, the force F is removed.
Find the time between the instant that the force is removed and the instant that Q comes to rest.
(4)
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